Abstract
Let \(G\) be a simple, connected graph with minimum degree \(\delta\ge2\), order \(n\), diameter \(\text{diam}(G)=d\) and degree distance \(D'(G)\). We prove that
\begin{eqnarray*}
D'(G)\le \begin{cases}
\frac{1}{4}nd\left(n-\frac{1}{2}\delta{d}\right)^2+O(n^3), \quad \mbox{if $G$ is triangle-free}\\
\frac{1}{4}nd\left(n-\frac{1}{5}(\delta^2-\delta+1){d}\right)^2+O(n^3), \quad
\mbox{if $G$ is $C_4$-free}.\end{cases}
\end{eqnarray*}
Although no construction has been found to show that the bounds are asymptotically tight, apart from improving known results in the literature, for triangle-free graphs the results confirm that an upper bound on the degree distance \(\frac{1}{32}n^4+O(n^3)\) conjectured by Dobrynin and Kochetova holds. This in conjunction with an infinite family of triangle-free graphs we construct in this paper that attain an upper bound on the degree distance, \(\frac{1}{8}nd\left(n-\frac{1}{2}\delta{d}\right)^2+O(n^3)\), give a guide for further research.
